Quantum measurement:can we recover evolution operator from measurement? Suppose there is a quantum system in state $|a\rangle$$\in$ H,an instrument designed to measure some physical quantity, which itself is in state $|b\rangle$ $\in$ H'. If we consider the bigger system "quantum system+the instrument",the state of this bigger system should be in $|a\rangle\otimes|b\rangle$. At t=0, the instrument started to measure the quantum system. At t=t',the instrument gives a result.
Now questions come:Can the measurement process described by an unitary operator U(t) of the bigger Hilbert space H$\otimes$H'?
Second, let the measurement corresponding to a self-adjoint operator P. If the answer of the above question is yes, Can we recover U(t) from P(or vice versa)?
Third, can t' equal to zero, which means instantaneous process?
 A: 
Can the measurement process be described by an unitary operator $U(t) $ of the bigger Hilbert space $H\otimes H'$?

There has been several attempts to derive the measurements process, that is the Born rule, from the unitary evolution. In particular, in this precise setting, by Zurek with his theory of decoherence (or quantum Darwinism). So, quantum Darwinism or decoherence is precisely an attempt to do that.

Second, let the measurement corresponding to a self-adjoint operator P. If the answer of the above question is yes, Can we recover U(t) from P (or vice versa)?

Even given decoherence theory, the measurement is not only determined by the observable (P) but also by the form of the interaction between the system and the environment (or laboratory). In other words the answer to this question is no. This is clear also given that $U(t)$ acts on the full Hilbert space while the observable measurement $P$, on the system alone. 

Third, can t' equal to zero, which means instantaneous process?

No process can really be instantaneous in physics. Instantaneous process is always an approximation, which may work very well if there is a suitable separation of time-scales. 
